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sqrt(2*sinx)-1
Plotting a function graph/ sqrt(2*sinx)-1

Graphing y = sqrt(2*sinx)-1

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The graph:

from to

Intersection points:

does show?

Piecewise:

The solution

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f(x) = \/ 2*sin(x)  - 1
f(x)=2sin⁡(x)−1f{\left(x \right)} = \sqrt{2 \sin{\left(x \right)}} - 1 
f = sqrt(2*sin(x)) - 1*1
The graph of the function
0-80-60-40-20204060802-2
The points of intersection with the X-axis coordinate
Graph of the function intersects the axis X at f = 0
so we need to solve the equation:
2sin⁡(x)−1=0\sqrt{2 \sin{\left(x \right)}} - 1 = 0
Solve this equation
The points of intersection with the axis X:

Analytical solution
x1=π6x_{1} = \frac{\pi}{6}
x2=5π6x_{2} = \frac{5 \pi}{6}
Numerical solution
x1=−97.9129710368819x_{1} = -97.9129710368819
x2=90.5825881785057x_{2} = 90.5825881785057
x3=27.7507351067098x_{3} = 27.7507351067098
x4=75.9218224617533x_{4} = 75.9218224617533
x5=−62.3082542961976x_{5} = -62.3082542961976
x6=34.0339204138894x_{6} = 34.0339204138894
x7=78.0162175641465x_{7} = 78.0162175641465
x8=−236.143047794833x_{8} = -236.143047794833
x9=−3.66519142918809x_{9} = -3.66519142918809
x10=−41.3643032722656x_{10} = -41.3643032722656
x11=94.7713783832921x_{11} = 94.7713783832921
x12=−74.8746249105567x_{12} = -74.8746249105567
x13=65.4498469497874x_{13} = 65.4498469497874
x14=−18.3259571459405x_{14} = -18.3259571459405
x15=−53.9306738866248x_{15} = -53.9306738866248
x16=−5.75958653158129x_{16} = -5.75958653158129
x17=−56.025068989018x_{17} = -56.025068989018
x18=25.6563400043166x_{18} = 25.6563400043166
x19=19.3731546971371x_{19} = 19.3731546971371
x20=−30.8923277602996x_{20} = -30.8923277602996
x21=−9.94837673636768x_{21} = -9.94837673636768
x22=−91.6297857297023x_{22} = -91.6297857297023
x23=163.886416762268x_{23} = 163.886416762268
x24=63.3554518473942x_{24} = 63.3554518473942
x25=2.61799387799149x_{25} = 2.61799387799149
x26=57.0722665402146x_{26} = 57.0722665402146
x27=−79.0634151153431x_{27} = -79.0634151153431
x28=−22.5147473507269x_{28} = -22.5147473507269
x29=−60.2138591938044x_{29} = -60.2138591938044
x30=−47.6474885794452x_{30} = -47.6474885794452
x31=46.6002910282486x_{31} = 46.6002910282486
x32=−49.7418836818384x_{32} = -49.7418836818384
x33=82.2050077689329x_{33} = 82.2050077689329
x34=−66.497044500984x_{34} = -66.497044500984
x35=15.1843644923507x_{35} = 15.1843644923507
x36=8.90117918517108x_{36} = 8.90117918517108
x37=84.2994028713261x_{37} = 84.2994028713261
x38=−100.007366139275x_{38} = -100.007366139275
x39=−81.1578102177363x_{39} = -81.1578102177363
x40=−93.7241808320955x_{40} = -93.7241808320955
x41=−28.7979326579064x_{41} = -28.7979326579064
x42=69.6386371545737x_{42} = 69.6386371545737
x43=6.80678408277789x_{43} = 6.80678408277789
x44=52.8834763354282x_{44} = 52.8834763354282
x45=21.4675497995303x_{45} = 21.4675497995303
x46=−68.5914396033772x_{46} = -68.5914396033772
x47=13.0899693899575x_{47} = 13.0899693899575
x48=−72.7802298081635x_{48} = -72.7802298081635
x49=−24.60914245312x_{49} = -24.60914245312
x50=−16.2315620435473x_{50} = -16.2315620435473
x51=31.9395253114962x_{51} = 31.9395253114962
x52=−85.3466004225227x_{52} = -85.3466004225227
x53=−37.1755130674792x_{53} = -37.1755130674792
x54=−12.0427718387609x_{54} = -12.0427718387609
x55=−43.4586983746588x_{55} = -43.4586983746588
x56=96.8657734856853x_{56} = 96.8657734856853
x57=38.2227106186758x_{57} = 38.2227106186758
x58=0.523598775598299x_{58} = 0.523598775598299
x59=59.1666616426078x_{59} = 59.1666616426078
x60=44.5058959258554x_{60} = 44.5058959258554
x61=−87.4409955249159x_{61} = -87.4409955249159
x62=88.4881930761125x_{62} = 88.4881930761125
x63=101.054563690472x_{63} = 101.054563690472
x64=50.789081233035x_{64} = 50.789081233035
x65=71.733032256967x_{65} = 71.733032256967
x66=−129.32889757278x_{66} = -129.32889757278
x67=40.317105721069x_{67} = 40.317105721069
x68=−35.081117965086x_{68} = -35.081117965086
The points of intersection with the Y axis coordinate
The graph crosses Y axis when x equals 0:
substitute x = 0 to sqrt(2*sin(x)) - 1*1.
(−1)1+2sin⁡(0)\left(-1\right) 1 + \sqrt{2 \sin{\left(0 \right)}}
The result:
f(0)=−1f{\left(0 \right)} = -1
The point:
(0, -1)
Extrema of the function
In order to find the extrema, we need to solve the equation
ddxf(x)=0\frac{d}{d x} f{\left(x \right)} = 0
(the derivative equals zero),
and the roots of this equation are the extrema of this function:
ddxf(x)=\frac{d}{d x} f{\left(x \right)} =
the first derivative
2sin⁡(x)cos⁡(x)2sin⁡(x)=0\frac{\sqrt{2} \sqrt{\sin{\left(x \right)}} \cos{\left(x \right)}}{2 \sin{\left(x \right)}} = 0
Solve this equation
The roots of this equation
x1=π2x_{1} = \frac{\pi}{2}
x2=3π2x_{2} = \frac{3 \pi}{2}
The values of the extrema at the points:
 pi         ___ 
(--, -1 + \/ 2 )
 2              

 3*pi           ___ 
(----, -1 + I*\/ 2 )
  2


Intervals of increase and decrease of the function:
Let's find intervals where the function increases and decreases, as well as minima and maxima of the function, for this let's look how the function behaves itself in the extremas and at the slightest deviation from:
The function has no minima
Maxima of the function at points:
x2=π2x_{2} = \frac{\pi}{2}
Decreasing at intervals
(−∞,π2]\left(-\infty, \frac{\pi}{2}\right]
Increasing at intervals
[π2,∞)\left[\frac{\pi}{2}, \infty\right)
Inflection points
Let's find the inflection points, we'll need to solve the equation for this
d2dx2f(x)=0\frac{d^{2}}{d x^{2}} f{\left(x \right)} = 0
(the second derivative equals zero),
the roots of this equation will be the inflection points for the specified function graph:
d2dx2f(x)=\frac{d^{2}}{d x^{2}} f{\left(x \right)} =
the second derivative
−2⋅(2sin⁡(x)+cos⁡2(x)sin⁡32(x))4=0- \frac{\sqrt{2} \cdot \left(2 \sqrt{\sin{\left(x \right)}} + \frac{\cos^{2}{\left(x \right)}}{\sin^{\frac{3}{2}}{\left(x \right)}}\right)}{4} = 0
Solve this equation
Solutions are not found,
maybe, the function has no inflections
Horizontal asymptotes
Let’s find horizontal asymptotes with help of the limits of this function at x->+oo and x->-oo
lim⁡x→−∞(2sin⁡(x)−1)=2⟨−1,1⟩−1\lim_{x \to -\infty}\left(\sqrt{2 \sin{\left(x \right)}} - 1\right) = \sqrt{2} \sqrt{\left\langle -1, 1\right\rangle} - 1
Let's take the limit
so,
equation of the horizontal asymptote on the left:
y=2⟨−1,1⟩−1y = \sqrt{2} \sqrt{\left\langle -1, 1\right\rangle} - 1
lim⁡x→∞(2sin⁡(x)−1)=2⟨−1,1⟩−1\lim_{x \to \infty}\left(\sqrt{2 \sin{\left(x \right)}} - 1\right) = \sqrt{2} \sqrt{\left\langle -1, 1\right\rangle} - 1
Let's take the limit
so,
equation of the horizontal asymptote on the right:
y=2⟨−1,1⟩−1y = \sqrt{2} \sqrt{\left\langle -1, 1\right\rangle} - 1
Inclined asymptotes
Inclined asymptote can be found by calculating the limit of sqrt(2*sin(x)) - 1*1, divided by x at x->+oo and x ->-oo
lim⁡x→−∞(2sin⁡(x)−1x)=0\lim_{x \to -\infty}\left(\frac{\sqrt{2 \sin{\left(x \right)}} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the right
lim⁡x→∞(2sin⁡(x)−1x)=0\lim_{x \to \infty}\left(\frac{\sqrt{2 \sin{\left(x \right)}} - 1}{x}\right) = 0
Let's take the limit
so,
inclined coincides with the horizontal asymptote on the left
Even and odd functions
Let's check, whether the function even or odd by using relations f = f(-x) и f = -f(-x).
So, check:
2sin⁡(x)−1=2−sin⁡(x)−1\sqrt{2 \sin{\left(x \right)}} - 1 = \sqrt{2} \sqrt{- \sin{\left(x \right)}} - 1
- No
2sin⁡(x)−1=−2−sin⁡(x)+1\sqrt{2 \sin{\left(x \right)}} - 1 = - \sqrt{2} \sqrt{- \sin{\left(x \right)}} + 1
- No
so, the function
not is
neither even, nor odd
The graph
Graphing y = sqrt(2*sinx)-1
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